Problem 25 - Prealgebra

Question

Find the least common multiple (LCM) of each pair of numbers or monomials. $$16 a^{2}, 14 a b$$

Step-by-Step Solution

Answer

The LCM of \(16a^2\) and \(14ab\) is \(112a^2b\).

Step 1: Factor the numbers and variables

First, we break down each term into its prime factors and variable components. For the number 16, the prime factorization is \(16 = 2^4\). For \(a^2\), it is already prime in terms of variables. The first term is therefore \(16a^2 = 2^4a^2\). For the number 14, the prime factorization is \(14 = 2 \times 7\). Combined with the variable \(ab\), the second term becomes \(14ab = 2 \times 7 \times a \times b\).

Step 2: Identify the maximum power for each factor

Now, identify the highest powers of all factors appearing in either term. The factors include 2, 7, \(a\), and \(b\). The highest power for the factor 2 is \(2^4\). For the factor 7, it is \(7^1\). For the factor \(a\), it is \(a^2\) and for \(b\), it is \(b^1\).

Step 3: Construct the LCM from the maximum powers

Using the maximum powers identified, the LCM is formed by multiplying them together: \(LCM = 2^4 \times 7^1 \times a^2 \times b^1\). When you multiply these together, the result is \(LCM = 16 \times 7 \times a^2 \times b = 112a^2b\).

Key Concepts

Prime Factorization

Prime factorization is a method used to express a number as a product of its prime factors, which are numbers that can only be divided by 1 and themselves. Understanding this concept is crucial when solving for the least common multiple (LCM) of numbers or monomials.

To perform prime factorization, repeatedly divide the number by the smallest prime numbers (such as 2, 3, 5, etc.) until only prime numbers are left. For example, for the number 16, the prime factorization is:

• 16 divided by 2 gives 8
• 8 divided by 2 gives 4
• 4 divided by 2 gives 2
• 2 divided by 2 gives 1

So, the factorization of 16 is expressed as:
\[ 16 = 2^4 \]

In cases involving variables, such as in the expression \( a^2 \), it's already considered as a prime factor in terms of variables. The idea is to identify all possible factors, including numbers and variables. The next step involves using these factors to find solutions like the LCM.

Monomials

A monomial is a mathematical expression that contains only one term - which can include numbers, variables, or a combination of both. It’s important to grasp the structure of monomials when dealing with operations like finding the LCM.

Becoming comfortable with monomials helps in operations such as combining them, factoring, and simplifying expressions. For instance, the monomial \(16a^2\) is formed by multiplying the number 16 and the variable \(a^2\). Another monomial example is \(14ab\), which involves both numbers and different variables.

When looking for the LCM of monomials, you analyze both the numerical and variable parts. This analysis requires:

• Breaking down numbers into their prime factors
• Considering the power of each variable within the monomials

Understanding monomials allows you to apply these steps effectively, ensuring a proper and accurate calculation of the LCM.

Power of Variables

Variables in mathematical expressions often come with exponents, which represent the power of the variable. Understanding the "power of variables" is essential because it plays a crucial role when calculating the LCM, especially in expressions with variables.

The power refers to how many times a number, called the base, is multiplied by itself. For example, in the monomial \( a^2 \), the variable \( a \) has a power of 2, meaning that \( a \) is multiplied by itself twice.

When determining the LCM of expressions that involve variables, it is critical to:

• Identify each variable involved in the terms
• Look for the highest power of each variable that appears in any of the expressions

These identifications ensure the construction of an LCM that accurately reflects the maximum influence of each variable. So for terms \( 16a^2 \) and \( 14ab \), the highest power of \( a \) is \( a^2 \) and of \( b \) is \( b^1 \). Such insights are pivotal in assembling the overall expression for the LCM.